Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual (...) ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject. (shrink)

Certain selected issues around the Gödelian anti-mechanist arguments which have received less attention are discussed.

Roughly speaking, computationalism says that cognition is computation, or that cognitive phenomena are explained by the agent‘s computations. The cognitive processes and behavior of agents are the explanandum. The computations performed by the agents‘ cognitive systems are the proposed explanans. Since the cognitive systems of biological organisms are their nervous 1 systems (plus or minus a bit), we may say that according to computationalism, the cognitive processes and behavior of organisms are explained by neural computations. Some people might prefer to (...) say that cognitive systems are ―realized‖ by nervous systems, and thus that—according to computationalism—cognitive computations are ―realized‖ by neural processes. In this paper, nothing hinges on the nature of the relation between cognitive systems and nervous systems, or between computations and neural processes. For present purposes, if a neural process realizes a computation, then that neural process is a computation. Thus, I will couch much of my discussion in terms of nervous systems and neural computation.1 Before proceeding, we should dispense with a possible red herring. Contrary to a common assumption, computationalism does not stand in opposition to connectionism. Connectionism, in the most general and common sense of the term, is the claim that cognitive phenomena are explained (at some level and at least in part) by the processes of neural networks. This is a truism, supported by most neuroscientific evidence. Everybody ought to be a connectionist in this general sense. The relevant question is, are neural processes computations? More precisely, are the neural processes to be found in the nervous systems of organisms computations? Computationalists say ―yes‖, anti-computationalists say ―no‖. This paper investigates whether any of the arguments on offer against computationalism have a chance at knocking it off.2 Ever since Warren McCulloch and Walter Pitts (1943) first proposed it, computationalism has been subjected to a wide range of objections.. (shrink)

It is argued, by use of specific examples, that mathematical understanding is something which cannot be modelled in terms of entirely computational procedures. Our conception of a natural number (a non-negative integer: 0, 1, 2, 3,…) is something which goes beyond any formulation in terms of computational rules. Our ability to perceive the properties of natural numbers depends upon our awareness, and represents just one of the many ways in which awareness provides an essential ingredient to our ability to understand. (...) There is no bar to the quality of understanding being the result of natural selection, but only so long as the physical laws contain a non-computational ingredient. (shrink)

In a paper published in 1975, Robert Jeroslow introduced the concept of an experimental logic as a generalization of ordinary formal systems such that theoremhood is a (or in practice ) rather than . These systems can be viewed as (rather crude) representations of axiomatic theories evolving stepwise over time. Similar ideas can be found in papers by Putnam (1965) and McCarthy and Shapiro (1987). The topic of the present article is a discussion of a suggestion by Allen Hazen, that (...) these experimental logics might provide an illuminating way of representing “the human mathematical mind”. This is done in the context of the well-known Lucas-Penrose thesis. Though we agree that Jeroslow's model has some merit in this context, and that the Lucas-Penrose arguments certainly are less than persuasive, some semi-technical doubts are raised concerning the alleged impact of experimental logics on the question of knowable self-consistency. (shrink)

Scientists routinely solve the problem of supplementing one’s store of variables with new theoretical posits that can explain the previously inexplicable. The banality of success at this task obscures a remarkable fact. Generating hypotheses that contain novel variables and accurately project over a limited amount of additional data is so difficult—the space of possibilities so vast—that succeeding through guesswork is overwhelmingly unlikely despite a very large number of attempts. And yet scientists do generate hypotheses of this sort in very few (...) tries. I argue that this poses a dilemma: either the long history of scientific success is a miracle, or there exists at least one method or algorithm for generating novel hypotheses with at least limited projectibility on the basis of what’s available to the scientist at a time, namely a set of observations, the history of past conjectures, and some prior theoretical commitments. In other words, either ordinary scientific success is miraculous or there exists a logic of discovery at the heart of actual scientific method. (shrink)

Roger Penrose formuló en 1989 un argumento contra la IA. Dicho argumento concluye que la explicación científico-matemática de la realidad es más amplia que la meramente computacional, porque existen ciertos aspectos de la realidad no-computables. Este artículo analiza dicho argumento y la discusión al respecto, para concluir que el tipo de argumento que quiere desarrollar Penrose está viciado de raíz, lo que impide llegar a las conclusiones deseadas. A la vez se sostiene la validez filosófica de sus conclusiones y se (...) apunta a la idea de no-localidad como consistente al hablar sobre la conciencia. (shrink)

This is a paper for a special issue of Semiotic Studies devoted to Stanislaw Krajewski’s paper. This paper gives some supplementary notes to Krajewski’s on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4–6 in Krajewski’s and classify some misunderstandings of Gödel’s incompleteness theorem related to AntiMechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel’s Disjunctive Thesis, Gödel’s Undemonstrability of Consistency Thesis and the definability (...) of natural numbers as in Section 7–8 in Krajewski’s, describing how recent advances bear on these issues. (shrink)

La cognition humaine paraît étroitement liée à la structure de l'espace et du temps relativement auxquels le corps, le geste, l'intelligibilité semblent devoir se déterminer. Pourtant, ce qui, après les approches physico-mathématiques de Galilée et de Newton, fut caractérisé par Kant comme formes de l'intuition sensible, n'a cessé au cours des siècles qui suivirent de se trouver remis en cause dans leur saisie première par les développements théoriques. En mathématiques d'abord, avec les géométries non-euclidiennes, en physique ensuite, où relativité générale (...) puis théories quantiques et critiques ont dû remanier profondément l'objectivité de ces concepts pour en faire des catégories, certes toujours aussi essentielles, mais de plus en plus contre-intuitives, et maintenant en biologie où la temporalité, notamment, et la causalité se révèlent largement différentes de celles de la physique. C'est ce que nous tentons de présenter et de discuter dans ce texte en vue d'en dégager la pertinence pour la cognition elle-même. (shrink)

Disertační práce se zabývá problematikou připisování myšlení jiným entitám, a to pomocí imitační hry navržené v roce 1950 britským filosofem Alanem Turingem. Jeho kritérium, známé v dějinách filosofie jako Turingův test, je podrobeno detailní analýze. Práce popisuje nejen původní námitky samotného Turinga, ale především pozdější diskuse v druhé polovině 20. století. Největší pozornost je věnována těmto kritikám: Lucasova matematická námitka využívající Gödelovu větu o neúplnosti, Searlův argument čínského pokoje konstatující nedostatečnost syntaxe pro sémantiku, Blockův návrh na použití brutální síly pro (...) řešení imitační hry, Frenchova teorie subkognitivních informací a Michieho skepticismus ohledně možnosti umělého vědomí. Závěr práce zachycuje současný stav recepce Turingova testu a představuje pokusy o jeho praktickou realizaci, například v každoroční soutěži o Loebnerovu cenu. Autor práce zastává názor, že ani po více než šedesáti letech od uveřejnění Turingova paradigmatického eseje stále neexistují žádné vážné důvody pro zamítnutí jeho tvrzení. Tradiční komputační funkcionalismus možná není ideální teorií vysvětlující činnost myslí a jako slibnější se může jevit vývoj v neurálních vědách, ale Turingův test je přesto užitečným a snad i jediným nástrojem pro detekci inteligence u lidmi vytvořených strojů. (shrink)

Albert Einstein once made the following remark about "the world of our sense experiences": "the fact that it is comprehensible is a miracle." (1936, p. 351) A few decades later, another physicist, Eugene Wigner, wondered about the unreasonable effectiveness of mathematics in the natural sciences, concluding his classic article thus: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" (1960, p. 14). (...) At least three factors are involved in Einstein's and Wigner's miracles: the physical world, mathematics, and human cognition. One way to relate these factors is to ask how the universe could possibly be structured in such a way that mathematics would be applicable to it, and we would be able to understand that application. This is roughly Wigner's question. Alternatively, the way of the mathematical naturalist is to argue that we abstract certain properties from the world, perhaps using our bodies and physical tools, thereby articulating basic mathematical concepts, which we continue building into the complex formal structures of mathematics. John Stuart Mill, Penelope Maddy, and Rafael Nuñez teach this strategy of cognitive abstraction, in very different manners. But what if the very concepts and basic principles of mathematics were built into our cognitive structure itself? Given such a cognitive a priori mathematical endowment, would the miracles of the link between world and cognition (Einstein) and mathematics and world (Wigner) not vanish, or at least significantly diminish? This is the stance of Stanislas Deheane and Elizabeth Brannon's 2011 anthology, following a venerable rationalist tradition including Plato and Immanuel Kant. (shrink)